Imaging Plasmon Hybridization in Metal Nanoparticle Aggregates with

Article pubs.acs.org/JPCC

Imaging Plasmon Hybridization in Metal Nanoparticle Aggregates with Electron Energy-Loss Spectroscopy Steven C. Quillin,† Charles Cherqui,† Nicholas P. Montoni,† Guoliang Li,‡ Jon P. Camden,*,‡ and David J. Masiello*,† †

Department of Chemistry, University of Washington, Seattle, Washington 98195, United States Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556, United States

‡

ABSTRACT: Electron energy-loss spectroscopy (EELS) provides detailed nanoscopic spatial and spectral information on plasmonic nanoparticles that cannot be discerned with far-field optical techniques. Here we demonstrate that EELS is capable of mapping the relative phases of individual localized surface plasmons that are hybridized within nanoparticle assemblies. Within the context of an effective plasmon oscillator model, we demonstrate the relationship between the self-induced backforce on the electron due to the plasmon and the EEL probability and use this to present a rubric for determining the relative phases of hybridized localized surface plasmons in EELS. Comparison between the analytical oscillator model, experiment, and numerical electrodynamics simulation is made across a variety of nanoparticle monomer, dimer, and trimer systems.

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INTRODUCTION Optical-frequency electromagnetic fields penetrate metal surfaces on the order of tens of nanometers as determined by the skin depth.1 Metal nanoparticles are unique in that applied fields can penetrate them entirely, facilitating the efficiency of their coupling to the radiation field. The largest amplitude optical excitations in metal nanoparticles are due to the collective displacements of their conduction-band electrons known as localized surface plasmon resonances (LSPRs). LSPRs have a number of interesting and unexpected optical properties, such as the ability to focus light below the diffraction limit. Such is useful in a variety of applications, including solar energy conversion,2,3 catalysis,4,5 enhanced molecular spectroscopy,6−10 and nanoscale heating.11,12 The specific characteristics of LSPRs are determined both by the composition and morphology of the nanostructure. They are labeled bright if they efficiently couple to the radiation field and dark if they do not. A variety of near-field optical methods have been used to probe the plasmonic properties of nanoparticles, but the spatial resolution of these methods is on the order of tens of nanometers,13,14 limiting their effectiveness. In contrast, electron energy-loss spectroscopy (EELS) in a scanning transmission electron microscope (STEM) has emerged as one of the leading techniques used in characterizing the plasmonic properties of individual and aggregate systems of nanoparticles, due in part to the fewnanometer-scale spatial resolution readily achieved in practice. The observables produced by STEM/EELS are dependent upon: (i) the spectral behavior of the STEM electron, which provides a near-continuum source of light with which to excite LSPR modes, and (ii) the location of the beam relative to the © 2016 American Chemical Society

target. Fixing the beam position produces an EEL probability spectrum over all possible loss energies from 0 eV (no loss) up to the kinetic energy of the STEM electron (typically ∼100 keV), while filtering a set of EEL spectra acquired at many beam positions at a specific loss-energy value produces an energy-filtered EEL map. The latter shows those regions in space where it is more or less probable for the STEM electron to lose energy and can be used to infer detailed information about the nanoscopic spatial profile of the particular LSPR excited at that loss-energy value. In this paper, we review a procedure for mapping LSPRs onto a collection of coupled mechanical oscillators and, within this context, derive an exact expression for the EEL probability in hybridized nanoparticle systems. We then use this expression to develop a rubric for determining the relative phases between the monomer LSPRs within a hybridized multiparticle assembly. For clarity and simplicity, our analysis is restricted to the dipole LSPR on each particle; however, extension to higher-order LSPRs is straightforward from the model presented. The analytical results are validated both by experiment as well as by EELS simulations using the MNPBEM method15 on equivalent nanoparticle systems. Dielectric data from ref 16 is used throughout. Special Issue: Richard P. Van Duyne Festschrift Received: March 1, 2016 Revised: April 14, 2016 Published: April 21, 2016 20852

DOI: 10.1021/acs.jpcc.6b02170 J. Phys. Chem. C 2016, 120, 20852−20859

Article

The Journal of Physical Chemistry C

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OSCILLATOR MODEL OF LSPRS In 1978, Lucas and co-workers17,18 established a method for mapping the dipole response of a set of interacting spherical nanoparticles onto a set of fictitious coupled mechanical oscillators with an effective mass m = e2/αspω2sp. Here, αsp = 3a3/(ϵ∞ + 2), a is the sphere radius, ωsp is the dipole LSPR frequency, and ε∞ is the static dielectric response of the metal’s core electrons; for simplicity, the background environment is chosen to be a vacuum. The plasmon’s effective mass is inversely proportional to the nanoparticle’s volume and is a measure of its polarizability: the smaller the mass the more polarizable, the larger the mass the less polarizable. Within the quasi-static limit, an assembly of coupled plasmonic nanoparticles located at positions ri and driven by the force F(ri,t) can be described by the effective Hamiltonian N

H=

∑ i=1

Pi2 1 + miωi2Q i2 − 2mi 2

∑ i≠j

determined by whether or not the force is resonant with the LSPR frequency. The optical absorption can be calculated by considering the average power absorbed by a plasmon over a cycle of oscillation T of the driving field ⟨Pabs(t )⟩T =

σabs(ω) =

where Qi = n̂iQi and Pi = n̂iPi represent the displacement coordinate and conjugate momentum, mi the effective mass, and ωi the resonance frequency of the ith dipole plasmon. LSPRs interact with each other pairwise via the near-field 3 dipole−dipole coupling tensor ΛNF ij = (3n̂in̂j−1ij)/|ri−rj| . The first two terms in eq 1 represent the internal energy of the LSPR, the third term accounts for the interparticle coupling, and the fourth term accounts for the application of an external force. From eq 1, a set of coupled equations of motion can be determined. Before moving on to examining the dynamics of coupled LSPRs, we first examine the dynamics of a single particle by setting N = 1 in eq 1. This yields F(t ) Q̈ + γ Q̇ + ωsp2Q = m

Eel(r, t ) =

Q(t ) =

∫−∞ dt′

e−γ(t − t ′)/2 ωsp2 − (γ /2)2

sin[

Eel(r, ω) =

ΓEELS(ω) = =

where g(t − t′) is the causal harmonic oscillator Green function. Its Fourier transform is proportional to the polarizability of the sphere via Im{g(ω)} = (4π/a3ω2sp)Im{α(ω)}. Light Excitation. The time evolution of a sphere’s dipole plasmon under the influence of a plane wave light field of force F(t) = −eE0cos(ωt) is

∞

dωℏω ΓEELS(ω)

(8)

⎡ ⎛ ωR ⎞ ⎤ 2eω ⎢ ⎛ ωR ⎞ ⎟⎟z ̂ − γLK1⎜⎜ ⎟⎟R̂ ⎥eiωz / v iK 0⎜⎜ 2 2 v γL ⎢⎣ ⎝ vγL ⎠ ⎝ vγL ⎠ ⎥⎦

(9)

|F(R , ω)|2 Im{α(ω)} e 2ℏπ

⎡ ⎤ 4e 2ω2 ⎢ 2⎛ ωR ⎞ 1 ⎛ ωR ⎞⎥ ⎟⎟ + 2 K 02⎜⎜ ⎟⎟ Im{α(ω)} K1 ⎜⎜ 4 2 ℏπv γL ⎢⎣ ⎝ vγL ⎠ γL ⎝ vγL ⎠⎥⎦

(10)

Comparing the expressions for ΓEELS to σabs in the case of plane wave excitation, we see that σabs depends on a single parameter, the frequency of the incoming light, while ΓEELS depends on three: frequency (corresponding to the energy loss), velocity, and beam location. The velocity of the electron is determined by the technical specifications of the STEM and can be considered a fixed parameter, typically about half the speed of light. At a fixed beam position, ΓEELS is a function of frequency alone, and eq 10 gives the probability that an electron will lose one quantum of energy to the LSPR, which, due to the resonance structure of α(ω), peaks at the LSPR frequency. Filtering a set of EEL spectra acquired over a region of interest at a specific loss energy value produces an EEL map, which will

−e E0cos(ωt − δ) −1

∫−∞ dtPabs(t ) = ∫0

where K0 and K1 are modified Bessel functions of the second kind. All together

F(t ′) − (γ /2) (t − t ′)] m 2

(3)

m (ωsp2 − ω 2)2 + (γω)2

(7)

where ΓEELS is a distribution function describing the probability per unit of transferred frequency for the electron to lose energy to the target19 over the characteristic interaction time T. ΓEELS is expressible in terms of the Fourier transform of the electron’s electric field

∫−∞ dt′g(t − t′) F(mt′)

Q(t ) =

− v t)2 + (R /γL)2 ]3/2

∞

t

=

−e(R + z − vt ) γL2[(z

T ⟨Pabs(t )⟩T =

(2)

ωsp2

(6)

that it carries. In this expression v = vẑ is the electron’s velocity, γL is the Lorentz contraction factor, and R is the distance in the impact plane from the plasmonic dipole to the electron beam. The EEL probability is related to the absorbed power in the following manner,

where an empirical friction term proportional to Q̇ has been added with damping rate γ to account for losses intrinsic to the metal. The formal solution is t

4πω Im{α(ω)} c

and is a function of frequency alone, as the spatial dependence encoded in the force is lost due to the delocalized nature of the plane wave at the nanoscale. STEM Electron-Beam Excitation. The force exerted by a uniformly moving relativistic electron within a STEM is dictated by the evanescent field

(1)

i=1

(5)

where σabs is the absorption cross section and I is the intensity of the light source. The cross section has the form

e2 Q ·ΛijNF·Q j 2 i

∑ F(ri, t )·Q i

T /2

∫−T /2 dt F(t )·Q̇ (t )

= σabs(ω)I

N

−

1 T

(4)

(ωγ/[ω2sp−ω2])

where δ = tan determines the relative phase between the force and the plasmon. From this expression, it is clear that the dynamics of the plasmon are determined by the applied force. However, the amplitude of its dynamics is 20853


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into collective modes. In contrast to a single sphere, the addition of a second spherical particle breaks the radial symmetry in the impact plane. To model the hybridization of two dipole LSPRs, we consider the Hamiltonian in eq 1 for N = 2. Ignoring the forcing for the moment, the subsystem of two coupled LSPRs can be diagonalized by introducing normal mode coordinates Q± = ∑λn̂λ±Qλ± with amplitudes

be important in the following in determining the degree of socalled “plasmon hybridization” in nanoparticle assemblies.20 Note that the loss probability is proportional to the magnitude of the force exerted by the electron on the plasmon or, via Newton’s second law, the self-induced back-force exerted by the plasmon on the electron. Quantization of H is straightforward and can be used to compute the EEL probabilities for higher quanta losses.18 Both ways of visualizing EELS data are displayed in Figure 1 for the case of a 30 nm diameter silver sphere in vacuum. Figure

Q +λ = cos θλQ 1λ + sin θλQ 2λ Q −λ = −sin θλQ 1λ + cos θλQ 2λ

(11)

where λ = x,y,z indicates directionality in space and θλ = (1/2) tan−1(2gλ12/μ[ω21 − ω22]) is the decoupling or mixing angle.21 Here μ = m1m2 is the geometric average of the effective λ masses of the particles and gλ12 = e2n̂λ1·ΛNF 12 ·n̂2 is the interparticle coupling strength, where n̂λi is the projection of the direction of the ith LSPR onto the direction λ. For the sake of generality, the LSPR effective masses and resonant frequencies are chosen to be different. The subscripts + and − refer to the in-phase and out-of-phase hybridized plasmon modes, respectively. There are analogous expressions for the conjugate momenta Pλ+ and Pλ−. Applying the coordinate transformation of eq 11 yields

Figure 1. EEL spectrum and map of a 30 nm diameter silver sphere. (A) EEL spectrum determined by Mie theory with the electron beam placed 5 nm from the surface of the sphere, indicated by the red bullet in panel D. The low energy peak (shaded in red) corresponds to the dipole plasmon resonance while the high energy shoulder is due to the excitation of higher order, multipolar plasmon modes. The dashed curve is the result from the oscillator model. (B) Forcing map from the oscillator model of the dipole plasmon mode. The circular ring of equal forcing is a result of the spherical symmetry of the system. (C) EELS maps corresponding to the dipole plasmon mode. The circular ring of equal EEL probability is again due to the particle’s symmetry. (D) TEM image of an experimentally measured silver sphere of 165 nm diameter. (E) Experimental EELS map of the dipole plasmon mode of sphere displayed in panel D. Both the model (C) and experimental (E) EEL maps have the same qualitative properties.

H0 =

P+2 1 P2 1 + μω+2Q +2 + − + μω−2Q −2 2μ 2 2μ 2

(12)

where the resonant frequencies for the hybridized modes are ω+λ =

ω12cos2 θλ + ω22 sin 2 θλ −

ω−λ =

ω12 sin 2 θλ + ω22cos2 θλ +

2g12λ μ 2g12λ μ

cos θλ sin θλ

cos θλ sin θλ

(13)

The new normal modes can be aligned along the axis of displacement resulting in an in-phase, collinear mode and an out-of-phase, anticollinear mode or they can be aligned perpendicular to the axis of displacement resulting in an inphase, parallel mode and an out-of-phase, antiparallel mode in the impact plane. An additional pair of degenerate parallel and antiparallel modes are oriented normal to the impact plane but will not be discussed as they are weakly excited due to the form of the electron’s electric field in that direction. In the basis of normal modes, the interaction between the electron and the LSPRs becomes

1A shows the EEL spectrum corresponding to a 5 nm impact parameter outside of the particle, based upon the model (dashed) and Mie theory (solid) for both the dipole and higher-order multipolar plasmons. The low energy peak (shaded in red) is due to the excitation of the dipole LSPR, while the high-energy shoulder is attributed to a collection of higher-order multipoles. EELS maps are displayed in Figure 1C,E. Figure 1B,C show contour plots of the magnitude of the force felt by the electron and ΓEELS, as determined from the Hamiltonian model, at the resonant frequency of the plasmon. Both are displayed in the plane perpendicular to the trajectory of the electron, called the impact plane. Figure 1E shows the experimentally measured EEL probability map for a silver sphere of 165 nm diameter (TEM image shown in Figure 1D), which has the same qualitative features. Because of the spherical symmetry of the nanoparticle, the induced dipole moment tracks the position of the electron beam, leading to a ring of equal EEL probability around its perimeter. The EEL probability drops to zero toward the center of the sphere because there the STEM electron polarizes the conduction electrons radially, which is incompatible with the dipole LSPR. This demonstrates the connection between the force applied on the electron due to the plasmon and the EEL probability.

⎡ ⎤ ⎛ m ⎞1/4 ⎛ m ⎞1/4 Hext = − ∑ (− e)⎢cos θλ⎜ 2 ⎟ Eλ(r1, t ) + sin θλ⎜ 1 ⎟ Eλ(r2, t )⎥Q +λ ⎢ ⎥ ⎝ m1 ⎠ ⎝ m2 ⎠ ⎣ ⎦ λ ⎡ ⎤ ⎛ m ⎞1/4 ⎛ m ⎞1/4 + (− e)⎢− sin θλ⎜ 2 ⎟ Eλ(r1, t ) + cos θλ⎜ 1 ⎟ Eλ(r2, t )⎥Q −λ ⎢ ⎥ ⎝ m1 ⎠ ⎝ m2 ⎠ ⎣ ⎦ = − [F+(t )·Q + + F−(t )·Q −]

(14)

where the new mixed forces F+ and F−, which arise from the fact that normal mode coordinates are linear combinations of the uncoupled coordinates, are implicitly defined. In analogy to the previous section, the associated ΓEELS for the hybridized system is

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NANOSPHERE DIMERS The behavior of nanoparticle aggregates becomes more complicated as a result of hybridization of individual LSPRs

ΓEELS(ω) =

∑ λ

|F+λ(ω)|2 2

e ℏπ

Im{α+λ(ω)} +

|F −λ(ω)|2 Im{α −λ(ω)} e 2ℏπ (15)

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The Journal of Physical Chemistry C where αλ+(ω) and αλ−(ω) are the effective polarizabilities of the normal modes oriented in the λ direction. From eq 14, it is evident that F+ and F− are superpositions of the forces acting on the individual uncoupled LSPRs, now weighted by factors proportional to the effective mass and mixing angle. For identical particles, these simplify to Fλ+ = −e(Eλ(r1,ω) + Eλ(r2,ω))/√2 and Fλ− = −e(−Eλ(r1,ω) + Eλ(r2,ω))/√2, resulting in a net zero force when Eλ(r1,ω) = −Eλ(r2,ω) for Fλ+ and Eλ(r1,ω) = Eλ(r2,ω) for Fλ−. Interestingly, when the electric field is due to a plane wave, each nanoparticle experiences the same spatially constant force, enforcing Eλ(r1,ω) = Eλ(r2,ω). This is in contrast to the electric field of a STEM electron, which has a high degree of spatial variation on the nanoscale, such that every point in space experiences a different force. This means that the electron beam imposes different selection rules depending upon its location in the impact plane. For example, it is impossible to drive the in-phase mode from the center of the dimer junction. Nanosphere Homodimer. This concept is further explored in Figure 2, where both force and EELS maps are

Figure 3. Simulated EEL spectra and maps of a nanosphere homodimer. Each sphere has a 30 nm diameter and is separated by 5 nm. (A) EEL spectra corresponding to three impact parameters of the electron beam, indicated by the colored bullets in the inset. (B) Simulated (left column) and experimental (right column) EEL maps of the hybridized dipole plasmon modes of the dimer system. The collinear mode (α) and the antiparallel mode (β) display nodal structure in the junction for both simulated and experimentally measured EEL maps. The parallel mode (γ) and the anticollinear mode (δ) have no nodal structure.

experiment. In order to observe all four relevant hybridized modes, three impact parameters are considered for the simulated EEL spectra shown in Figure 3A. The first beam location, labeled by the red bullet, is oriented along the axis of displacement of the two spheres on the exterior side and excites two of the dimer modes: mode α and mode γ. The second beam location, labeled by the blue bullet, is oriented to the side of one of the spheres in the dimer, off of the axis of displacement, and excites three modes: mode α, mode β, and mode γ. The final beam position, labeled by the green bullet, is oriented directly between and equidistant from the two spheres and excites two modes: mode γ and mode δ. Each hybridized mode in this system can be identified by considering the nodal structure observed in the EELS maps and areas of higher EEL probability. Two modes, α and β, contain nodal structure in the junction, a feature that only occurs for the collinear and antiparallel modes. Mode α shows regions of higher EEL probability near the exterior of the spheres along the axis of displacement. Mode β shows regions of higher EEL probability along the sides of the nanospheres off of the axis of displacement. The two modes lacking nodal structure, γ and δ, can likewise be identified by observing the location of higher EEL probability, as they must either be anticollinear or parallel modes. Mode γ displays higher EEL probability along the sides of the nanospheres off of the axis of displacement identifying it as the parallel mode. Mode δ displays higher EEL probability in the junction around the axis of displacement identifying it as the anticollinear mode. Based upon this nodal structure, the four hybridized modes of the sphere dimer can be identified and categorized. Extending these ideas, the effects of asymmetry in the dimer system may now be analyzed. Nanosphere Heterodimer. Further examination of the forcing term described by eq 14 shows that in addition to the spectral and spatial behavior of the field of the electron beam, parameters inherent to the dimer, such as effective mass, resonance frequency, and interparticle coupling act as weighting factors for the force applied on each nanosphere, thereby affecting ΓEELS. In this section, the effect of breaking the symmetry of the dimer is examined in two ways: (i) demanding that both LSPR frequencies are the same (i.e., ω1 = ω2) while varying the effective mass of one particle, and (ii) varying the

Figure 2. Forcing and EEL maps of the nanosphere homodimer based upon the Hamiltonian model. Regions of zero force and zero EEL probability are displayed in white. (A−D) Forcing maps displaying the forcing of the collinear (A), anticollinear (B), parallel (C), and antiparallel (D) modes. Each point in space corresponds to the magnitude of the force exerted by the electron beam located at that position. In the collinear (A) and antiparallel (D) mode force maps, a nodal line appears between the two particles equidistant from their adjacent surfaces indicating zero net force. The locations of zero forcing translate directly into nodes in the EEL maps. (A′−D′) EEL maps for the collinear (A′), anticollinear (B′), parallel (C′), and antiparallel (D′) modes computed from the Hamiltonian model of the hybridized dimer system. Nodal structure appears in the EEL maps of the collinear and antiparallel modes colocated with the lines of zero force in panels A and D.

depicted for the four hybridized modes of interest of the nanosphere homodimer based upon the above Hamiltonian model. Nodes are spatially colocated in both the forcing and EELS maps for the in-phase, collinear and the out-of-phase, antiparallel modes of the system. The node bisecting the junction of the dimer is a result of the force on the mode netting zero, leaving the mode undriven for impact parameters along this line. Experimental validation of the presented dimer model can be found in Figure 3 together with numerical electrodynamics simulations of the electron probe. While the experimental data is taken from a pair of spherical particles each of 165 nm diameter, the qualitative nature of the modes does not change from those considered in the model. A high degree of correlation is evident between the model, simulation, and 20855


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Figure 4. EEL maps showing the effects of changing the effective mass or resonance frequency for one particle in the nanosphere dimer for the parallel, antiparallel, collinear, and anticollinear plasmon modes. Regions of zero EEL probability are displayed in white. The central column, outlined in red, corresponds to identical particles (m1 = m2 and ω1 = ω2), while panels to the left of this show the effect of detuning the left particle’s resonance frequency from that of the right, and panels to the right show the result of changing the effective mass ratio by increasing the effective mass of the left particle. Localization is observed in all four modes with the introduction of heterogeneity but localize on the same particle as the effective mass is changed and on different particles as the resonance frequency is changed.

LSPR frequency of one particle while fixing their effective masses to be the same (i.e., m1 = m2). These cases are explored in Figure 4 for the parallel and antiparallel modes (upper panels) and the collinear and anticollinear modes (lower panels). The central column corresponds to a system of identical nanoparticles (m1 = m2 and ω1 = ω2). Moving right from this column displays the effects of increasing the effective mass of the left particle, leading to localization of the EEL probability on the particle with a lower effective mass, regardless of which mode is excited. This effect is a consequence of the fact that the particle with the lower effective mass has a higher polarizability, leading to a stronger induced electric field than the particle of effective greater mass. How the system behaves for equal effective masses but different resonance frequencies is explored by moving left from the center column. This also leads to localization but for a different reason. This localization is entirely due to the mixing of modes and is instead a signature of Fano-like interferences.22−24 This can be verified by building EELS maps for each set of ω+ and ω− modes and looking for a flip in plasmon localization. This is in contrast to the case of degenerate plasmon resonances and differing effective masses which is entirely determined by the polarizability of each individual nanoparticle and therefore shows no change in localization.

for the nanorod.25−27 Along its longitudinal axis, the nanorod is more polarizable and has a smaller effective mass, while along its short axes, it is less polarizable and has a larger effective mass. Depending upon its aspect ratio, the longitudinal and transverse plasmons can be energetically split to not overlap, with the longitudinal plasmon lying lower in energy than the transverse. For these reasons, EEL maps of nanospheres and nanorods show a stark difference. While the nanosphere has circular rings of equal probability in its EEL maps, the nanorod has lobes of high EEL probability at its tips and a node at its center. Aligning two rods along their longitudinal axes results in a system with hybridized plasmon modes analogous to the collinear modes of the sphere dimer and exhibits the same nodal structure or lack thereof. Conversely, separating the two rods in parallel results in hybridized plasmon modes analogous to the parallel modes of the sphere dimer. While the presented Hamiltonian model can be extended to describe prolate spheroids which approximate the nanorod, we have chosen only to interpret numerical electrodynamics simulations, the results of which are displayed in Figure 5. Nanorod Trimers. Rod systems become more interesting when a third rod is added. By arranging the rods in the appropriate geometry, it is evident that EELS maps not only measure relative phase but also show signatures of the excitation of degenerate modes. Two rod trimers are considered in the following: one in which the rods are arranged in a zigzag pattern and one where they are arranged on the sides of a triangle. Figure 6 displays the case of three rods arranged in a zigzag pattern. Figure 6A shows the EEL spectrum for such a trimer

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NANOROD SYSTEMS As polarizability depends on the geometry of a nanoparticle, and the effective mass depends on the polarizability, introducing anisotropy into the geometry of a nanoparticle will change the effective masses of the LSPRs. Such is the case 20856


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signifying that it is the net dipole mode. In mode β, rather than seeing nodal structure resulting from pairs of collinear dipoles, the EEL probability is more localized to the outer particles in both junctions. This is because the peak in the spectrum corresponds to two possible orientations of the dipole plasmon on the central particle. Because both are equally polarizable, the EEL map encodes the superposition of these two modes. Mode γ does not contain nodal structure or asymmetry in the EEL probability and is the maximally out-of-phase dipolar plasmon mode. The EEL spectrum and corresponding maps of the triangular nanorod trimer system are shown in Figure 7. The spectrum,

Figure 7. Simulated EEL spectrum and maps of a silver nanorod trimer arranged in an equilateral triangle. Each rod is 80 nm in length and 10 nm in diameter. (A) EEL spectrum of the trimer computed at the beam position denoted by the red bullet in the inset. (B) EEL maps associated with the peaks α′ and β′ in panel A. Mode α′ contains nodes in every junction, meaning that each junction contains a headto-tail configuration of dipoles. Mode β′ does not display any nodes in the junctions. Although it is impossible for every junction to have a head-to-head configuration, the electron beam’s ability to force degenerate modes leads to a superposition of the EELS maps for both orientations. The double-sided arrow displayed on the rightmost rod represents the dipole orientation for the pair of degenerate modes.

Figure 5. EEL maps of the nanorod monomer and dimer. (A) The EEL map of a single nanorod displays lobes of high EEL probability at the ends and a node in the center. (B) The collinear mode (top) of the longitudinal rod dimer has lobes of high loss probability at the outer ends and a nodal line in the junction. In contrast, the anticollinear mode (bottom) has a lobe of high EEL probability in the junction. (C) The antiparallel mode (top) of the parallel rod dimer has lobes of high EEL probability at the ends with a nodal line in the junction between particles. The parallel mode (bottom) shows similar behavior but does not have a nodal line in the junction.

shown in Figure 7A, contains two distinct peaks with the corresponding maps shown in Figure 7B. For the mode labeled α′, nodes appear in all three junctions, corresponding to a headto-tail configuration. Such head-to-tail configurations in sphere trimer systems have been known to exhibit magnetic plasmon behavior.23,28−33 The mode labeled β′ does not display any nodal structure in the junctions. This is counterintuitive, as it is impossible for every dipole to be oriented in a head-to-head configuration. This, once again, is a result of the electron beam driving two degenerate modes simultaneously. The double sided arrow displayed represents the dipole orientation for each degenerate mode.

Figure 6. Simulated EEL spectrum and maps of a silver nanorod trimer arranged in a zigzag configuration with right angles formed between each particle. Each rod is 80 nm in length and 10 nm in diameter. (A) EEL spectrum of the nanorod trimer computed at the beam position denoted by the red bullet in the inset. (B) EEL maps associated with the peaks α, β, and γ in panel A. Mode α corresponds to the net plasmon dipole, where each nanorod dipole is oriented head-to-tail. This is characterized in the EEL map by an enhanced nodal behavior in the junctions. Mode β corresponds to one junction having a head-to-tail configuration and the other having a separate head-to-head configuration. Due to the electron’s ability to force degenerate modes, a node is not observed. The double-sided arrow displayed on the central rod represents the dipole orientation for each degenerate mode. Mode γ is characterized by an all head-to-head configuration of the nanorod dipoles.

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METHODS EELS Experiments. EELS experiments are carried out in a monochromated Carl Zeiss LIBRA 200MC (S)TEM operated at 200 kV. All EEL maps are acquired with a convergence semiangle of 9 mrad, a collection semiangle of 12 mrad, and a dispersion of 29 meV per channel. The measured energy resolution (defined as the full width at half-maximum of the zero-loss peak) is 150 meV. For the single particle EELS map acquisition, a region of interest with 30 × 29 pixel spectra (1 pixel ∼8.1 nm × 8.1 nm) is defined over the entire silver nanoparticle. However, for the dimer EEL map acquisition, a region of interest with 45 × 23 pixel spectra (1 pixel ∼9.9 nm × 9.9 nm) is created. To generate the plasmon mode maps, the background from all pixel spectra is removed through the reflected-tail model routine using the Gatan Digital Micrograph software, and then normalize pixel-by-pixel by the zero-loss

and has three distinct peaks. Figure 6B shows the EEL maps corresponding to each peak. Mode α displays nodes in the junctions that extend toward the middle of the outer particles, 20857


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intensity. Finally, by plotting the EEL probability in designated energy slices, EEL maps are generated.

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CONCLUSIONS EEL maps can be interpreted as a measure of the self-induced back-force on the electron due to the excitation of LSPRs. In this paper, we show not only that the relative phase between coupled LSPRs can be inferred through the emergence of nodal structure in EEL maps but also that EEL maps yield information about superpositions of degenerate modes. To support these claims, we have developed a Hamiltonian formalism to model STEM/EELS measurements on hybridized plasmonic nanoparticle assemblies and compare it to experiment and numerical electrodynamics simulations. A variety of monomer, dimer, and trimer systems are discussed. Particular emphasis is placed on comparing heterodimers of mixed polarizability to those of differing resonant frequencies, showing that care must be taken when interpreting plasmon localization in EELS maps. In particular, we show that degenerate systems of mixed polarizability also show localization, but it is not related to the mixing of modes usually associated with plasmon localization in Fano-like systems. We apply this reasoning to the numerical study of dimer and trimer hybridized nanorod systems and find character attributed to the superposition of both modes in the case of degenerate trimer plasmons. This work provides a rubric for assigning the relative phases of each monomer within hybridized nanoparticle aggregates, thereby determining the character of the modes under study.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: (574) 631-1059. *E-mail: [email protected]. Phone: (206) 5435579. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Basic Energy Sciences, under award number DE-SC0010536 (J.P.C., G.L.), a Notre Dame Energy postdoctoral fellowship (G.L.), the National Science Foundation’s CAREER program under award number CHE-1253775 (D.J.M.), NSF XSEDE resources under award number PHY-130045 (D.J.M.), and the State of Washington through the University of Washington Clean Energy Institute (S.C.Q.).

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REFERENCES

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Imaging Plasmon Hybridization in Metal Nanoparticle Aggregates with

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